Minimum value of f(x)=(x^p)/(p)+(x^(-q))/(q) subjected to 1/p+1/q=1 and p>1 Although I have solved it using derivative. But did not understand how can i solve without derivative, Help Required, Thanks

maredilunavy 2022-09-14 Answered
minimum value of f ( x ) without the use of derivative
Although I have solved it using derivative. But did not understand how can i solve
without derivative, Help Required, Thanks
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Answers (2)

altaryjny94
Answered 2022-09-15 Author has 6 answers
Use Young's inequality.
f ( x ) = x p p + x q q x 1 x = 1.
Since f ( 1 ) = 1, the minimum is 1
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Krish Crosby
Answered 2022-09-16 Author has 3 answers
By AM-GM:
x p p + x q q 1 p + 1 q + x + 1 x 2 1 p + 1 q = 1
Since x p p x + p 1 0 and x q q x + q 1 0
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( x y ) 2 z 2 ( x + y ) 2 z 2 ÷ x y + z x + y z
I have been solving several of this type by:
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3.- Removing factors of 1.
In this case, i'm trying to find a way of solving this without expanding the hole thing, but i have not found the way jet.
I start by expanding both square binomials of the dividend.
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The problem is that i don't know what to do now. I know that the result is:
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